Lindelöf hypothesis

E518478

conjecture in analytic number theory mathematical conjecture unproven conjecture

The Lindelöf hypothesis is an unproven conjecture in analytic number theory about the growth rate of the Riemann zeta function along the critical line, with deep implications for the distribution of prime numbers.

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All labels observed (3)

Label	Occurrences
Lindelöf hypothesis canonical	3
Lindelöf hypothesis (background and consequences)	1
Lindelöf hypothesis for automorphic L-functions	1

How this entity was disambiguated

This entity first appeared as the object of triple T5425834 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Lindelöf hypothesis
Context triple: [Ernst Lindelöf, notableFor, Lindelöf hypothesis]

A. Riemann hypothesis
The Riemann hypothesis is a famous unsolved conjecture in number theory asserting that all nontrivial zeros of the Riemann zeta function lie on a critical line in the complex plane, with deep implications for the distribution of prime numbers.
B. Ramanujan–Petersson conjecture
The Ramanujan–Petersson conjecture is a fundamental statement in number theory and the theory of modular forms that predicts strong bounds on the Fourier coefficients of modular cusp forms, with deep connections to automorphic forms and the Langlands program.
C. Hilbert–Pólya conjecture
The Hilbert–Pólya conjecture is an unproven idea in number theory suggesting that the nontrivial zeros of the Riemann zeta function correspond to eigenvalues of a suitable self-adjoint operator, offering a potential spectral approach to proving the Riemann hypothesis.
D. generalized Riemann hypothesis
The generalized Riemann hypothesis is a major unproven conjecture in number theory asserting that the nontrivial zeros of all Dirichlet L-functions lie on a critical line in the complex plane, extending the classical Riemann hypothesis.
E. Riemann zeta function
The Riemann zeta function is a complex-valued function central to analytic number theory, whose properties—especially the distribution of its zeros—are deeply connected to the distribution of prime numbers.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Lindelöf hypothesis
Target entity description: The Lindelöf hypothesis is an unproven conjecture in analytic number theory about the growth rate of the Riemann zeta function along the critical line, with deep implications for the distribution of prime numbers.

A. Riemann hypothesis
The Riemann hypothesis is a famous unsolved conjecture in number theory asserting that all nontrivial zeros of the Riemann zeta function lie on a critical line in the complex plane, with deep implications for the distribution of prime numbers.
B. Ramanujan–Petersson conjecture
The Ramanujan–Petersson conjecture is a fundamental statement in number theory and the theory of modular forms that predicts strong bounds on the Fourier coefficients of modular cusp forms, with deep connections to automorphic forms and the Langlands program.
C. Hilbert–Pólya conjecture
The Hilbert–Pólya conjecture is an unproven idea in number theory suggesting that the nontrivial zeros of the Riemann zeta function correspond to eigenvalues of a suitable self-adjoint operator, offering a potential spectral approach to proving the Riemann hypothesis.
D. generalized Riemann hypothesis
The generalized Riemann hypothesis is a major unproven conjecture in number theory asserting that the nontrivial zeros of all Dirichlet L-functions lie on a critical line in the complex plane, extending the classical Riemann hypothesis.
E. Riemann zeta function
The Riemann zeta function is a complex-valued function central to analytic number theory, whose properties—especially the distribution of its zeros—are deeply connected to the distribution of prime numbers.
F. None of above. chosen

Statements (44)

Predicate	Object
instanceOf	conjecture in analytic number theory ⓘ mathematical conjecture ⓘ unproven conjecture ⓘ
concerns	Riemann zeta function ⓘ asymptotic behavior of the Riemann zeta function ⓘ growth of the Riemann zeta function on the critical line ⓘ
concernsProperty	order of growth of \|ζ(1/2+it)\| ⓘ
domain	critical line Re(s) = 1/2 of the complex plane ⓘ
field	analytic number theory ⓘ number theory ⓘ
hasConsequence	bounds for error terms in prime-counting functions ⓘ results on the distribution of primes in arithmetic progressions ⓘ results on the distribution of primes in short intervals ⓘ
hasGeneralization	Lindelöf hypothesis for Dirichlet L-functions ⓘ Lindelöf hypothesis for automorphic L-functions ⓘ linked to: Lindelöf hypothesis
hasHeuristicSupportFrom	random matrix theory models of the zeta function ⓘ
hasImplicationFor	distribution of zeros of the Riemann zeta function ⓘ error term in the prime number theorem ⓘ size of gaps between primes ⓘ
hasOpenStatusIn	Clay Millennium Problem context (via relation to Riemann Hypothesis) ⓘ
implies	strong bounds on the growth of the Riemann zeta function on the critical line ⓘ subpolynomial growth of the Riemann zeta function on the critical line ⓘ upper bounds for moments of the Riemann zeta function on the critical line ⓘ
isConnectedTo	moment conjectures for the Riemann zeta function ⓘ subconvexity problems for L-functions ⓘ
isDiscussedIn	analytic number theory textbooks ⓘ research literature on the Riemann zeta function ⓘ
isNotKnownToBeEquivalentTo	Riemann Hypothesis ⓘ linked to: Riemann hypothesis
isPartOf	classical problems about the Riemann zeta function ⓘ
isStrongerThan	trivial bounds for \|ζ(1/2+it)\| ⓘ
isSubjectOf	ongoing research in analytic number theory ⓘ
isWeakerThan	Riemann Hypothesis ⓘ linked to: Riemann hypothesis
mathematicalObject	statement about complex-valued function growth ⓘ
motivatedBy	understanding fine distribution of primes ⓘ
namedAfter	Ernst Leonard Lindelöf ⓘ linked to: Ernst Lindelöf
relatedTo	Dirichlet L-functions ⓘ Riemann Hypothesis ⓘ linked to: Riemann hypothesis distribution of prime numbers ⓘ generalized Lindelöf hypothesis ⓘ
statedIn	early 20th century ⓘ
status	open problem ⓘ
typeOfBound	upper bound on \|ζ(1/2+it)\| as t → ∞ ⓘ
usedIn	analytic estimates in prime number theory ⓘ study of zeros of L-functions ⓘ

How these facts were elicited

Referenced by (5)

Full triples — surface form annotated when it differs from this entity's canonical label.

Ernst Lindelöf → notableFor → Lindelöf hypothesis ⓘ

Ernst Lindelöf → hasConceptNamedAfter → Lindelöf hypothesis ⓘ

A. Ivić, The Riemann Zeta-Function → coversTopic → Lindelöf hypothesis (background and consequences) ⓘ

linked to: Lindelöf hypothesis

generalized Riemann hypothesis → relatedTo → Lindelöf hypothesis ⓘ

Lindelöf hypothesis → hasGeneralization → Lindelöf hypothesis for automorphic L-functions ⓘ